Hilbert–Burch theorem

In mathematics, the Hilbert–Burch theorem describes the structure of some free resolutions of a quotient of a local or graded ring in the case that the quotient has projective dimension 2. Hilbert (1890) proved a version of this theorem for polynomial rings, and Burch (1968, p.944) proved a more general version. Several other authors later rediscovered and published variations of this theorem. Eisenbud (1995, theorem 20.15) gives a statement and proof.

Statement

If R is a local ring with an ideal I and

is a free resolution of the R-module R/I, then m = n – 1 and the ideal I is aJ where a is a non zero divisor of R and J is the depth 2 ideal generated by the determinants of the minors of size m of the matrix of the map from R^m to Rⁿ.

References

David Hilbert

David Hilbert (German: [ˈdaːvɪt ˈhɪlbɐt]; 23 January 1862 – 14 February 1943) was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry. He also formulated the theory of Hilbert spaces, one of the foundations of functional analysis.

Hilbert adopted and warmly defended Georg Cantor's set theory and transfinite numbers. A famous example of his leadership in mathematics is his 1900 presentation of a collection of problems that set the course for much of the mathematical research of the 20th century.

Hilbert and his students contributed significantly to establishing rigor and developed important tools used in modern mathematical physics. Hilbert is known as one of the founders of proof theory and mathematical logic, as well as for being among the first to distinguish between mathematics and metamathematics.

Hilbert (name)

Hilbert is both a Germanic masculine given name and surname. Notable people with the name include:

Given name:

Surname:

Hilbert (crater)

Hilbert is a lunar crater that is located on the far side of the Moon, just past the southeast limb. It lies just beyond the region of the surface that is occasionally brought into view due to libration, and so this feature cannot be observed directly from the Earth.

The crater is attached to the southeast rim of the walled plain Pasteur, a formation nearly half as large again as Hilbert. To the southeast of Hilbert is the smaller crater Alden, while Backlund lies to the west-northwest.

Much of the outer rim of Hilbert remains relatively intact, although it is heavily eroded in places particularly in the south. The interior is relatively flat, but has been deeply gouged by several small craters. There is a short range of central ridges offset to the west of the interior midpoint. Just to the west of the ridges is Hilbert W, a crater with a smaller crater overlying the western rim, giving it a pear-like shape. Hilbert H is a circular, bowl-shaped crater in the east part of the floor. Next to the northern rim is Hilbert Y. There are also many lesser craterlets scattered about the interior.

Theorem

In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems—and generally accepted statements, such as axioms. A theorem is a logical consequence of the axioms. The proof of a mathematical theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is often interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific law, which is experimental.

Many mathematical theorems are conditional statements. In this case, the proof deduces the conclusion from conditions called hypotheses or premises. In light of the interpretation of proof as justification of truth, the conclusion is often viewed as a necessary consequence of the hypotheses, namely, that the conclusion is true in case the hypotheses are true, without any further assumptions. However, the conditional could be interpreted differently in certain deductive systems, depending on the meanings assigned to the derivation rules and the conditional symbol.

Teorema (film)

Teorema is a 1968 Italian art-house film written and directed by Pier Paolo Pasolini and starring Terence Stamp, Laura Betti, Silvana Mangano, Massimo Girotti, and Anne Wiazemsky. It was the first time Pasolini worked primarily with professional actors. In this film, an upper-class Milanese family is introduced to, and then abandoned by, a divine force. Two prevalent motifs are the desert and the timelessness of divinity.

Plot

A mysterious figure known only as "The Visitor" appears in the lives of a typical bourgeois Italian family. His arrival is heralded at the gates of the family's Milanese estate by an arm-flapping postman. The enigmatic stranger soon engages in sexual affairs with all members of the household: the devoutly religious maid, the sensitive son, the sexually repressed mother, the timid daughter and, finally, the tormented father. The stranger gives unstintingly of himself, asking nothing in return. He stops the passionate maid from committing suicide with a gas hose and tenderly consoles her; he befriends and sleeps with the frightened son, soothing his doubts and anxiety and endowing him with confidence; he becomes emotionally intimate with the overprotected daughter, removing her childish innocence about men; he seduces the bored and dissatisfied mother, giving her sexual joy and fulfillment; he cares for and comforts the despondent and suffering father, who has fallen ill.

Dynkin system

A Dynkin system, named after Eugene Dynkin, is a collection of subsets of another universal set satisfying a set of axioms weaker than those of σ-algebra. Dynkin systems are sometimes referred to as λ-systems (Dynkin himself used this term) or d-system. These set families have applications in measure theory and probability.

The primary relevance of λ-systems are their use in applications of the π-λ theorem.

Definitions

Let Ω be a nonempty set, and let be a collection of subsets of Ω (i.e., is a subset of the power set of Ω). Then is a Dynkin system if

Ω ∈ ,

if A, B ∈ and A ⊆ B, then B \ A ∈ ,

if A₁, A₂, A₃, ... is a sequence of subsets in and A_n ⊆ A_n+1 for all n ≥ 1, then .

Equivalently, is a Dynkin system if

Ω ∈ ,

if A₁, A₂, A₃, ... is a sequence of subsets in such that A_i ∩ A_j = Ø for all i ≠ j, then .

The second definition is generally preferred as it usually is easier to check.

An important fact is that a Dynkin system which is also a π-system (i.e., closed under finite intersection) is a σ-algebra. This can be verified by noting that condition 3 and closure under finite intersection implies closure under countable unions.